Solve the systems of equations by elimination
3x-5y = 29
2x+3y = -6
We can solve the system of equations by elimination. To do this, we will multiply both sides of the second equation by 5 and both sides of the first equation by 3 to get the same coefficient for y in both equations:
3(3x-5y) = 3(29)
5(2x+3y) = 5(-6)
This simplifies to:
9x - 15y = 87
10x + 15y = -30
Now we can add the two equations together:
(9x - 15y) + (10x + 15y) = 87 + (-30)
This simplifies to:
19x = 57
Dividing both sides by 19, we get:
x = 3
Now we can substitute x = 3 into one of the original equations to solve for y. Let's use the first equation:
3(3) - 5y = 29
Simplifying:
9 - 5y = 29
Subtracting 9 from both sides:
-5y = 20
Dividing both sides by -5, we get:
y = -4
Therefore, the solution to the system of equations is x = 3 and y = -4.
To solve the system of equations by elimination, we can eliminate one variable by adding or subtracting the equations. Let's solve it step-by-step:
Step 1: Multiply the first equation by 2 and the second equation by 3 to make the coefficients of x in both equations equal:
(2)(3x - 5y) = (2)(29)
(3)(2x + 3y) = (3)(-6)
Simplifying, we get:
6x - 10y = 58
6x + 9y = -18
Step 2: Subtract the second equation from the first equation to eliminate the x variable:
(6x - 10y) - (6x + 9y) = 58 - (-18)
6x - 10y - 6x - 9y = 58 + 18
Simplifying, we get:
-19y = 76
Step 3: Divide both sides of the equation by -19 to solve for y:
-19y / -19 = 76 / -19
y = -4
Step 4: Substitute the value of y back into one of the original equations, let's use the first equation:
3x - 5*(-4) = 29
3x + 20 = 29
3x = 29 - 20
3x = 9
Step 5: Divide both sides of the equation by 3 to solve for x:
3x / 3 = 9 / 3
x = 3
Therefore, the solution to the system of equations is x = 3 and y = -4.
To solve the systems of equations by elimination, we need to eliminate one variable by adding or subtracting the two equations. The goal is to create a new equation in which one variable is eliminated, so we can solve for the remaining variable.
Let's proceed step by step:
1. Multiply the first equation by 2 and the second equation by 3 to make the coefficients of x in both equations the same:
Equation 1: 2(3x - 5y) = 2(29) => 6x - 10y = 58
Equation 2: 3(2x + 3y) = 3(-6) => 6x + 9y = -18
2. Now, we can eliminate x by subtracting the second equation (6x + 9y = -18) from the first equation (6x - 10y = 58). This will eliminate the variable x:
(6x - 10y) - (6x + 9y) = 58 - (-18)
Simplifying the equation gives: -19y = 76
Divide both sides of the equation by -19: y = -4
3. Substitute the value of y = -4 into either of the original equations and solve for x. Let's use the first equation, 3x - 5y = 29:
3x - 5(-4) = 29
3x + 20 = 29
3x = 9
Divide both sides of the equation by 3: x = 3
So, the solution to the system of equations is x = 3 and y = -4.